Snapping not working at all
Well, I compared with another file, setting-for-setting, until I found the answer. It turns out that in the 3d-view header (in edit mode) there is a button for "Proportional Editing Mode", and when I set that to "disabled" then snapping started working again. I have no idea what I did to enable that (probably accidentally pressed some obscure key combo). I searched all over the Interweb for this issue and although I found a bunch of folks with the problem, nobody offered this "solution". I hope this helps the next poor soul that has this issue.